We had a small but loyal crowd that included a three-year old and an eight-year old. The three-year old was charming, as all three-year olds are, and today she answered all yes/no questions in the affirmative. She and I talked about shapes and eggs and money. It was good times.

But I really got to get into the head of the eight-year old.

We discussed the grapefruit page below, and the unsolved mystery of whether there are exactly six grapefruit—the ones we can see directly—or more than that with at least one hiding underneath, possibly reflected in the surface of the bowl.

We moved on to the next page, which is where the real fun began.

My eight-year old conversation partner looked carefully, thought for a while, and announced that there must have been more than six grapefruit on the previous page because there are more than six on this page.

I asked, “How do you know?” and it turns out he was visually pairing the grapefruit halves on this page. He used his fingers to show me the pairs he made, but he was having trouble keeping track of their number. So when he came out with more than six pairs of grapefruit halves on this page, that meant there must have been more grapefruit in the bowl.

We flipped pages back and forth several times while sorting this out, and he finally concluded that there were six grapefruit on both pages. Children rarely have math tasks that connect this way, but they expect that the tasks should connect. It was delightful to watch this expectation play out.

Next up was the avocado page.

He thought for a bit and decided there were “seven point five avocados”. I thought I knew how he knew—same as the grapefruit—but I asked to be sure, and I was wrong.

“Three fives is fifteen, and then divide by two.”

It took a few more exchanges to extract that dividing by two makes sense here because there should be half as many whole avocados as there are half-avocados. Of course this is brilliant and important mathematics, and it arose in the context of making sense of a meaningful counting situation. Also notable is that three fives was a fact he retrieved quickly while three fours (of grapefruit halves) did not seem to occur to him.

The lesson here is that children are brilliant. They build math out of their everyday experiences, and when you offer them opportunities they apply the math they know to make further sense of their worlds.

Another lesson is that my new book—titled How Many?—is out. The best price and free shipping are at Stenhouse.com. If you read it with children, please report back and maybe leave a review at Amazon.

Tabitha (9 years old) is keenly attuned to the temperatures these days, as subzero air temperatures or wind chills mean indoor recess. Being a child of great physical energy, indoor recess is not ideal.

We have an indoor/outdoor thermometer on our kitchen table, which she checks several times a day. Yesterday evening before doing the dishes together, she checks the thermometer.

Tabitha (9 years old): It’s 1 below.

Me: What was it this morning? Five degrees?

T: Four

Me: Crazy. So it’s colder now.

T: Yeah.

Me: How much colder?

T: Five below

Me: How do you know? Is it because 4 + 1, or did you count?

T: Neither.

Me: Oh! Now I have to hear it!

T: Well…Four minus four is zero, then it’s one less, so it’s five.

Me: So one more than four less…er…one less than four….no….

[we laugh]

T: It’s one more because it’s one less!

So what do we learn?

This conversation reminded me very much of a game I used to play with Griffin (who is now 12 years old) on cold winter mornings. In both cases, the children naturally developed a strategy using zero as a stopping point in making comparisons.

The thing I especially love about this story is that Tabitha expresses a complicated relationship that is crystal clear to her: “One more because it’s one less.” Expanded out, she’s saying that “The difference between -1 and 4 is bigger than the difference between 0 and 4—the difference is bigger by 1 because -1 is one unit further from 4 than 0 is.”

She can express this complicated idea because it is her own.

If I tried to tell her that this is how subtraction with negative numbers works, she would definitely pronounce my ideas confusing—whether they were expressed in the language of 9-year-olds or the language of mathematicians.

I cannot tell her these things and have them be meaningful. What I can do is ask how much colder it is now than it was this morning.

Starting the conversation

Move to Minnesota.

I’m kidding.

You can buy a Celsius thermometer, though.

You can make comparisons more generally, both asking your child how she knows, and talking about how you think about it. How many more full cups in the muffin tin than empty ones? How many more fork than spoons? How many more adults on the bus than children (or vice versa)? How many more quarters than dimes in the change bowl?

Summer has arrived in Minnesota, and that means we alternate between warm days where we open the windows and run the ceiling fan, and hot days where we close everything up and run the air conditioning (a luxury, btw, that our 1928-built home only got about five years back).

Tabitha is naturally curious about how the ceiling fan works. In case you don’t have experience with them, or yours works differently from ours, here are the basics: There is a switch on the wall—just like a light switch—that powers the fan. Then there is a chain hanging from the fan itself that affects the speed. There are four settings controlled by the chain—High, Medium, Low, and Off.

By the time this conversation takes place, Tabitha and I have already explored a variety of ceiling fan questions, such as If the fan is off, should you pull the chain to turn it on, or head over to the light switch? and How many times can I pull the chain before my parents tell me to stop playing with the fan?

On this day, I ask Tabitha to flip the wall switch to turn on the fan, which she does. She then starts to stand up on the couch to reach the chain. I ask why.

Tabitha (9 years old): I want to see if it’s on high.

Me: But how will you know? If you pull the chain it will slow down, but that’s what it always does. So how will you know whether it was on high to begin with?

T: Well, it doesn’t always slow down, otherwise how would it ever be on high?

So What Do We Learn?

There is some very deep math going on here.

Tabitha and I are playing with properties of modular arithmetic, but she (and you) don’t need to know the specifics. Things that go in cycles are all examples of this kind of math.

The classic example is time. You could say that later times have bigger numbers. 4 is later than 3; 12 is later than 9. This is just like my claim that every pull of the chain slows down the fan. Both of these claims are only sort of true. Three in the afternoon is later than 11 in the morning, despite having a smaller number. If the fan is on low and you pull the chain twice, it’ll be on high.

People study these things in great depth in the field of Modern Algebra, and the ideas are useful in all sorts of places.

Starting the Conversation

Play with a ceiling fan. Talk about staying up all night. Notice together that weird things happen when the fan is in the off position, and at midnight and noon. Wonder aloud whether 12 o’clock is like zero (and if not, what is?)

Play around with basic facts in this ceiling fan environment. If it’s on high, how many pulls to turn it off? If it’s on low, how many to get it to medium? I just pulled the chain three times and now it’s on low—where was it before? Etc. Challenge the child; have the child challenge you.

I am writing a book. In the process of doing this, I come across homework assignments that parents find frustrating, and that they share on social media. These almost always get me thinking, and they frequently lead to math talks with my children.

This past weekend was one such instance.

Talking Math with Your Kids is not a place to hash out the details of whether this is a well written question, or whether this was an appropriate homework assignment for this child. We can discuss that on Twitter if you like, or through my About/Contact page.

Talking Math with Your Kids is about taking opportunities to have math conversations with our children. In that spirit, I share the conversation we had in our house.

Out of the blue, I asked Tabitha (7 years old) if I could ask her a math question. It was maybe Saturday afternoon. We had nothing special going on.

Me: Tabitha, can I ask you a math question?

Tabitha (7 years old): Yes.

Me: If I have eight things, and seven of them are in one hand, how many are in the other?

T: That’s not even a math question! That’s too easy!

Me: OK. But will you answer it anyway?

T: One.

Me: OK. What if I had five in one hand?

T: And you still had 8?

Me: Yeah.

She spent a few moments thinking.

T: Three.

I had a couple other questions, which I asked and she answered. The next day, I realized that I didn’t know how she knew that second one.

She was getting ready to brush her teeth on Sunday evening when I asked whether she remembered the previous day’s conversation. She did.

Me: How did you know it was three?

T: I counted.

Me: Like this? Five, then six, seven, eight?

T: Yeah. And that’s three. But actually, I kind of already had it memorized.

Me: Oh yeah? How did you memorize it?

T: Huh?

Me: Did you try to memorize it? When I want to memorize a phone number because someone told it to me and I don’t have a pen handy, I say it over and over to myself. Did you do that with 5 + 3?

T: No! I just have counted it out a lot of times.

—

Now, I should also mention that I asked Tabitha, If I had 8 things, and 8 of them were in one hand, how many would be in the other? She replied Zero without much hesitation. This If I have this many in one hand, how many are in the other formulation is probably less clumsy than the If this is one part, what is the other part? formulation on the original worksheet. But the intention is the same.

So What Do We Learn?

The kind of problem Tabitha and I were working with is called Part-Part-Whole. For young children, this is different from the standard “takeaway” problem because there is no “taking away”. I didn’t eat, lose, destroy or give away any of my eight things in these problems—I just have some in one hand and some in the other.

Because Part-Part-Whole involves a different way of thinking, it’s a good idea to practice some of these problems. It helps children to build a better understanding of addition and subtraction relationships if they see all the various ways these relationships appear in their worlds.

Tabitha herself pointed out an important principle of Talking Math with Your Kids: Many things that you hope to remember, you can remember by encountering them frequently. Tabitha has never sat down with flash cards to memorize her single-digit addition facts. Yet she is in second grade and is starting to feel confident with them.

She and I talked about familiarity—how maybe learning 5 + 3 is a little like learning the name of someone you see in your neighborhood. You don’t recognize the person as being the same person the first few times you see them. But eventually, if you see them frequently enough, you do recognize them, and you might introduce yourself. Pretty soon, you know their name. And if you just can’t seem to remember it? That’s when it’s time to drill yourself. That’s when you repeat the name over and over and over.

Starting the Conversation

Ask the questions I did. This is an easy conversation to have. If your child isn’t confident with addition and subtraction facts, ask about six in one hand instead of jumping to five in one hand.

More broadly, look for Part-Part-Whole opportunities to talk about. This is an important interpretation of subtraction, and one that is often neglected. Examples include apples (Our fruit bowl has 8 apples—5 are red, how many are green?), pets (There are 8 pets on our block—5 are cats, the rest are dogs. How many dogs?), et cetera.

A propos of nothing one day, I ask Griffin (9 years old at the time, finishing up fourth grade) a question.

Me: Griff, imagine you are baking cookies and you need cup of sugar, but you only have a cup measure. How would you get cup?

He thinks about this for a moment.

Griffin (9 years old): You put cup of whatever you’re measuring.

Me: Sugar.

G: Does it matter?

Me: No. I suppose not.

The conversation could end here and I would be delighted. But it does not end here.

G: You put that into the bowl, then you fill the cup halfway and put that in.

Me: And that’s cup?

G: Yes.

Me: How do you know?

G: Because is a half, and then half of a half.

Me: Yeah. That is what you just described. How do you know that that’s right?

G: Like a square. If you shade in half of it, and then half of what’s left, that’s the same as shading of it.

So What Do We Learn?

One question division helps answer is how many of this are in that? My question of Griffin asked how many halves are in three-fourths? This is a division question.

Griffin may not know that it is a division question. That is fine. He is thinking about a specific example of how many of this are in that? This will lead to good things further down the line.

That he sees “sugar” as a non-essential detail of the story is lovely. This will serve him well.

Griffin’s mental image for this task is a common one. He can see three fourths of a square in his mind, and he can see that this is the same as one-and-a-half halves of a square.

Finally, we learn (because I am about to tell you) that this scenario could never really happen when baking in our home. I have an awesome set of measuring cups (pictured below): , , , , , 1 and . (A friend—and friend of the project—has pledged to donate her cup measure to the Talking Math with Your Kids cause.)

Starting the Conversation

There are so many ways to raise the question how many of this are in that? Measure each other in inches, wonder how many feet tall that is. Count your quarters, wonder how many dollars that is. Repeat with nickels, or dimes. Bake a batch of cookies using only the cup measure.

It has been a long, busy semester for me in my community college work. Many interesting and productive projects, lots of interesting and challenging teaching problems.

But I am tired. Wiped out and exhausted.

So I devised a plan the other evening when Tabitha needed to finish her first-grade math homework. I would lie on the daybed on the porch with my eyes closed while she worked at the adjacent table. I could answer any questions she might have without opening my eyes. (Seriously, parents—you may mock me, but can you honestly say you haven’t tried something similar?)

This plan worked beautifully for about five minutes.

She was working through some addition facts when it occurred to me that I had never asked her one of my favorite math questions. So I wrote the following in my notebook.

Me: What goes in the box?

Tabitha (7 years old): (reading aloud in a mumble to herself) Eight plus four is…

Hey! This doesn’t make any sense!

Me: Why not?

T: 8 plus 4 is something, then plus 5?

Me: What does the equal sign mean?

T: Is. Like 2 plus 2 is 4.

Me: What about this? Would it make sense to write 2 plus 2 equals 3 plus 1?

T: No!

I let it go and we move on with our evening.

Later on, though, after putting on jammies but before toothbrushing, I follow up.

Me: Tabitha, I want to ask you a follow up math question.

T: OK.

Me: Does it make sense to say 2 plus 2 is the same as 3 plus 1?

T: Yes! Of course! Easy!

Me: Can I let you in on a little secret?

T: A secret secret? Or not really a secret?

Me: Not really a secret. But something you might not know.

T: [rolls eyes] OK.

Me: The equal sign means “is the same as”.

T: Of course! I know that!

Me: But that means it would be OK to say that 2 plus 2 equals 3 plus 1.

T: Oh.

So what do we learn?

This is kind of a big deal.

We train children to think that the equal sign means and now write the answer. Arithmetic worksheets reinforce this idea. Calculators do too. (What button do you press to perform a computation on a typical calculator? The equal sign!)

But doing algebra requires that we understand the equal sign to mean is the same as or has the same value as.

Tabitha is in first grade, though, so she has lots of time to learn the correct meaning, right?

Sadly, older students in U.S. schools do worse on the task I gave Tabitha than younger ones do.

The good news is this: If we are aware that children may develop the wrong idea about the equal sign, it is easy to help them to get it right.

You can follow Tabitha’s and my adventures in equality in the coming weeks.

Starting the conversation

If you have a school-aged child of any age, pose that task above. No judgment. No hints. Report your results below. It’ll be fun!

Tabitha received a Twister game for her recent birthday (7 years old!) She enjoys a version of the game in which one person spins and the other follows instructions until, as Tabitha puts it with much delight, the cookie crumbles.

The players switch roles for the next round. No score is kept.

She wants to play a round one recent Sunday evening. I have been writing, so I have her set it up in the kitchen while I finish up.

She comes back to me with questions.

Tabitha (7 years old): Daddy! What’s six plus six plus six plus six?

Me: Wait. How many sixes?

T: Four.

Me: Twenty-four.

T: Yes! I counted them right!

Me: Huh?

She takes me into the kitchen to show me the Twister board.

T: See? One, two, three, four, five, six.

She is counting the green dots in one row.

T: Then one, two, three, four

She is counting the rows.

Me: So four sixes is 24. Nice. Can I show you something cool? It’s also six fours. See? One, two, three, four.

I am counting the dots in one column—each a different color.

Me: Then one, two, three, four, five, six.

I am counting the columns.

Me: So four sixes and six fours are the same.

T: Like the dominoes.

She is referring to a recent homework assignment in which dominoes were used to demonstrate that 6+4 is the same as 4+6, and that this is true as a general principle about addition.

So what do we learn?

Rows and columns are fun, fun, fun.

Malke Rosenfeld of Math in Your Feet reminds me regularly that children love to play in structured space. She uses blue tape on the floor for her math/dance lessons and has noticed that children love to play freely in and around the spaces created by the tape (seriously: click that link, have a read and then go buy some painter’s tape!). The same thing is true for the Twister board. It creates a structured space for Tabitha to explore at a scale that allows her to use her whole body. That’s a good time for a seven-year-old.

But children don’t always notice the rows and columns in an arrangement like the Twister board. They need to learn to notice it. This is an important step on the path to learning multiplication. The fact that our conversation began with “What is 6+6+6+6 ?” tells me that Tabitha notices the rows and the columns. She knows that the answer to 6+6+6+6 should be the same as her count. By introducing the language of “four sixes” and “six fours”, I am trying to help her notice the multiplication structure underlying her ideas.

Starting the conversation

Arrange things in rows and columns. When you do, the whole thing is called an array.

Point out arrays in the world. Count the number in each row together, and count the number of rows. Notice together whether the numbers switch if you count the number in each column and count the columns. Does eight rows of six become six columns of eight? Does this happen for all numbers?

Jennifer is in the kitchen baking chocolate chip cookies when her son Ian (8 years old) wanders in and observes her methods. She has put three balls of cookie dough in a row, two balls of dough in the next row, and is beginning a new row.

Ian (8 years old): Are you going to put three in the next row?

Mom: Yep.

Ian: And then two in the last row?

Mom: Yep…How many cookies are on the tray?

Simulated cookie dough. Shout out to anyone who can ID the actual substance in this photo.

Ian: Ten.

Mom: How do you know that?

Ian: Three plus three is six, and two plus two is four, and six plus four is ten.

Mom: Hmm….my brain immediately puts the three and two together to make 5 and then adds the 5s together.

Mom: The recipe days it makes 5 dozen cookies. How many is that?

Ian: So that’s 5 12′s?

Mom: Yes.

Ian: 36? No…24 plus… No, wait. 60.

Mom: Ok, I made a double batch, so how many is that?

Ian: 120

Mom: And if there’s 10 on a tray, how many trays of cookies will that be?

Ian: 12

Mom: I have three cookie sheets, so how many times will I have to put each tray in the oven?

Ian: 12 divided by 3 is 4 – four times.

So what do we learn?

What I love about this conversation is that every question is an authentic one that someone baking cookies might consider along the way. I love that Jennifer keeps asking questions until she hits one that forces Ian to think, and I love that she offers Ian a different way to view the cookies on the tray (2 fives instead of 6+4). This last bit sends an important message—that math ideas are something we talk about, not just memorized facts.

Most of the time when people think about the math involved in baking, it’s the fractions. Fractions of a cup and of a teaspoon are fine. But we don’t actually do much math with them. If I need , I usually measure 3 cups and then use the cup measure. It’s counting the whole way. This is good, and it’s useful for helping children become accustomed to the relative sizes of fractions, and to the language surrounding them. But there isn’t as much mathematical thinking going on as when Jennifer asks Ian how many cookies are in 5 dozen, or to say how he knows how many cookies are on a tray with a 3-2-3-2 pattern.

Starting the conversation

Baking together is a great opportunity for asking howmany? questions of various forms. Ask your child to put things in rows, or to count things that already are. Guess how many chocolate chips are in each cookie, and then in the whole batch. Compare to the expected number of raisins in an oatmeal cookie.

All along the way, listen to your child’s thinking and offer your own ideas. Make it a conversation.

Milk has been on sale at our local gas station/convenience store. Griffin and I walked up there the other day to buy some milk. Two percent milk for the kids and me; skim for Mommy.

Me: Griff, the milk we just bought was $5.50 for two gallons. How much was each gallon?

Griffin (9 years old): With tax included? Or not included? I don’t do tax problems.

Note: I weep for the loss of 4% and 5% sales tax rates. They were so easy to compute mentally, and such a nice introduction to the financial world for elementary age children. Minnesota’s sales tax rate is presently %. The city of Saint Paul tacks on another half percentage point. I don’t even bother computing sales tax mentally any more.

Me: No worries about taxes. There is no tax on milk.

G: OK. Two twenty-five. Er…no that’d be $4.50.

So…

$2.75!

Me: How did you do that?

G: Well, I thought it would be $2.25, but that’s half for $4.50, so there’s an extra dollar. So I split that dollar in half, which is 50 cents, put that with the $2.25, which is $2.75.

Me: Nice. I could see that thinking in your first answer; when you said $2.25. I was curious whether you used that first wrong answer or started over from scratch.

When I thought about it, I did it differently. I thought that half of $5.00 is $2.50, then I need to add half of 50 cents. Same answer, though. $2.75.

This thinking is not closely related to the standard long division algorithm. One of the big challenges in school curriculum is relating mental math strategies such as Griffin’s to efficient algorithms that are more useful for complicated computations. I have a few resources parents may find helpful over at Sophia.org.

Starting the conversation

Anytime you find yourself wondering about such things, ask your child to think along with you. I wanted to know whether the gas station price for a gallon of milk was a good one. This required me knowing what the price was for each gallon. Not a hard problem for me, but I had to think for a moment. So then I asked Griffin. Do the same at the grocery store, the convenience store, the hardware store; anyplace where things are priced in groups.

If your kid needs a challenge, ask about gasoline. I paid x for y gallons yesterday. How much per gallon? This one will likely require estimation skills!

It is written by a math teacher who converses with his niece (who is 7 years old) about rectangles and multiplication. As an example, the rectangle below shows that 6×3 is 18. Or is it that 3×6 is 18? That becomes the focus of part of the conversation.

The girls’ parents look on as the discussion unfolds.

At one point, the math teacher stops the mother who is trying to intervene to help the child see that 4×3 is the same as 3×4. And this leads to the lovely sentence in the blog post:

I understand that it is not obvious to non-teachers that not every encounter with mathematics needs to reach “fruition.”

What he means by this is that children can learn from thinking about math, even if they don’t end up with the right answer, and even if they do not experience the full story (here, that multiplication is commutative, which means AxB=BxA for all possible numbers).

Finally, non-math teacher parents may be interested to learn that—consistent with Fawn’s observation—a regular piece of feedback I get from math teachers on my writing here is how impressed they are by my ability to not worry about Tabitha and Griffin getting right answers.